@article{Zhang2023, 
author = {Huiping Zhang and Wangjin Yao},
title = {Three solutions for a three-point boundary value problem with instantaneous and non-instantaneous impulses},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {9},
pages = {21312-21328},
keywords = {three-point boundary value problem, variational method, critical points theorem, instantaneous impulse, non-instantaneous impulse},
url = {https://www.sciopen.com/article/10.3934/math.20231086},
doi = {10.3934/math.20231086},
abstract = {In this paper, we consider the multiplicity of solutions for the following three-point boundary value problem of second-order    p-Laplacian differential equations with instantaneous and non-instantaneous impulses:          {                                        −            (            ρ            (            t            )                          Φ                              p                                      (                          u              ′                        (            t            )            )                          )              ′                        +            g            (            t            )                          Φ                              p                                      (            u            (            t            )            )            =            λ                          f                              j                                      (            t            ,            u            (            t            )            )            ,                        t            ∈            (                          s                              j                                      ,                          t                              j                +                1                                      ]            ,                        j            =            0            ,            1            ,            .            .            .            ,            m            ,                                                Δ            (            ρ            (                          t                              j                                      )                          Φ                              p                                      (                          u              ′                        (                          t                              j                                      )            )            )            =            μ                          I                              j                                      (            u            (                          t                              j                                      )            )            ,                        j            =            1            ,            2            ,            .            .            .            ,            m            ,                                                ρ            (            t            )                          Φ                              p                                      (                          u              ′                        (            t            )            )            =            ρ            (                          t                              j                                            +                                      )                          Φ                              p                                      (                          u              ′                        (                          t                              j                                            +                                      )            )            ,                        t            ∈            (                          t                              j                                      ,                          s                              j                                      ]            ,                        j            =            1            ,            2            ,            .            .            .            ,            m            ,                                                ρ            (                          s                              j                                            +                                      )                          Φ                              p                                      (                          u              ′                        (                          s                              j                                            +                                      )            )            =            ρ            (                          s                              j                                            −                                      )                          Φ                              p                                      (                          u              ′                        (                          s                              j                                            −                                      )            )            ,                        j            =            1            ,            2            ,            .            .            .            ,            m            ,                                                u            (            0            )            =            0            ,                        u            (            1            )            =            ζ            u            (            η            )            ,                                  where        Φ          p        (  u  )  :=      |    u            |              p      −      2        u  ,    p  >  1  ,    0  =      s          0        <      t          1        <      s          1        <      t          2        <  .  .  .  <      s                  m                  1                      <      t                  m                  1                    +      1        =  η  <  .  .  .  <      s          m        <      t          m      +      1        =  1  ,    ζ  >  0  ,    0  <  η  <  1,    Δ  (  ρ  (      t          j        )      Φ          p        (      u    ′    (      t          j        )  )  )  =  ρ  (      t          j              +        )      Φ          p        (      u    ′    (      t          j              +        )  )  −  ρ  (      t          j              −        )      Φ          p        (      u    ′    (      t          j              −        )  ) for        u    ′    (      t          j              ±        )  =      lim          t      →              t                  j                          ±                          u    ′    (  t  ),    j  =  1  ,  2  ,  .  .  .  ,  m, and        f          j        ∈  C  (  (      s          j        ,      t          j      +      1        ]  ×      R    ,      R    ),        I          j        ∈  C  (      R    ,      R    ).    λ  ∈  (  0  ,  +  ∞  ),    μ  ∈      R   are two parameters.    ρ  (  t  )  ≥  1,    1  ≤  g  (  t  )  ≤  c for    t  ∈  (      s          j        ,      t          j      +      1        ],    ρ  (  t  )  ,    g  (  t  )  ∈      L          p        [  0  ,  1  ], and    c is a positive constant. By using variational methods and the critical points theorems of Bonanno-Marano and Ricceri, the existence of at least three classical solutions is obtained. In addition, several examples are presented to illustrate our main results.}
}